CHALLENGE PROBLEMS

Updated: Wed, Mar 20, 1996

1. Classification of the units in the ring Q[sqrt(2)]: + or - (sqrt(2)+1)^n, n=0, 1,-1,2,-2,...
2. p.381, #13.
3. If R is an integral domain and if every nonzero element in R which is not a unit can be represented as a product of irreducibles in R, does it follow that R must necessarily satisfy the ascending chain condition (see page 393)?
4. Prove that arctan(x) is irrational if x>0 is rational not 1. (Extend the class proof for arctan(1/2)). You may use the classification of prime elements in Z[i].) -Proved in class.
5. Prove that there are infinitely many primes of the form 4k+3 and 4k+1. - Proved in class.
6. Characterize the integers which can be represented as the sum of two squares of integers.
7. Prove that there are infinitely many primes of the form 6k+1 and 6k+5.
8. Prove that if an Abelian group G has two elements of finite orders m, n, then there is also an element in G whose order is [m,n] (the least common multiple of m and n). (Note: The above statement is true even without the assumption of G being Abelian.)
9. Prove that for odd primes p>2 the regular p^2-gon is not constructible. (The case p=3 has been discussed in class.)